Jingyu Huang

Stochastic heat equation with rough dependence in space

This paper studies the nonlinear one-dimensional stochastic heat equation driven by a Gaussian noise which is white in time and which has the covariance of a fractional Brownian motion with Hurst parameter 1/4<H<1/2 in the space variable. The existence and uniqueness of the solution u are proved assuming the nonlinear coefficient is differentiable with a Lipschitz derivative and vanishes at 0. In the case of a multiplicative noise, that is the linear equation, we derive the Wiener chaos expansion of the solution and a Feynman-Kac formula for the moments of the solution. These results allow us to establish sharp lower and upper asymptotic bounds for the moments of the solution.

Space-time fractional diffusions in Gaussian noisy environment

This paper studies the linear stochastic partial differential equation of fractional orders both in time and space variables , where is a general Gaussian noise and , . The existence and uniqueness of the solution, the moment bounds of the solution are obtained by using the fundamental solutions of the corresponding deterministic counterpart represented by the Fox H-functions. Along the way, we obtain some new properties of the fundamental solutions.

Large time asymptotics for the parabolic Anderson model driven by spatially correlated noise

In this paper we study the linear stochastic heat equation, also known as parabolic Anderson model, in multidimension driven by a Gaussian noise which is white in time and it has a correlated spatial covariance. Examples of such covariance include the Riesz kernel in any dimension and the covariance of the fractional Brownian motion with Hurst parameter

Smoothness of the joint density for spatially homogeneous SPDEs

H\in (\frac 14, \frac 12]

Large time asymptotics for the parabolic Anderson model driven by space and time correlated noise

in dimension one. First we establish the existence of a unique mild solution and we derive a Feynman-Kac formula for its moments using a family of independent Brownian bridges and assuming a general integrability condition on the initial data. In the second part of the paper we compute Lyapunov exponents, lower and upper exponential growth indices in terms of a variational quantity. The last part of the paper is devoted to study the phase transition property of the Anderson model.

On the intermittency front of stochastic heat equation driven by colored noises

In this paper we consider a general class of second order stochastic partial differential equations on

On stochastic heat equation with measure initial data

\mathbb{R}^d

Chaos expansion of 2D parabolic Anderson model

driven by a Gaussian noise which is white in time and it has a homogeneous spatial covariance. Using the techniques of Malliavin calculus we derive the smoothness of the density of the solution at a fixed number of points

Stochastic heat equations with general multiplicative Gaussian noises: Hölder continuity and intermittency

(t,x_1), \dots, (t,x_n)

On Hölder continuity of the solution of stochastic wave equations in dimension three

,